NiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material]

Transcription

1 NiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material] S. K. Panda, I. dasgupta, E. Şaşıoğlu, S. Blügel, and D. D. Sarma Partial DOS, Orbital projected band structure (Fatband) and Fermi Surface DOS (States/eV-Atom-Spin) - Ni-d S-p Energy (ev) Figure S: Partial densities of states of Ni-d and S-p for both the spin channel.

2 Majority Spin Channel Ni d Γ S p K M Γ A L H -8 Γ A K M Γ A L H A L H A Minority Spin Channel Ni d Γ S p K M Γ A L H -8 Γ A K M Γ A Figure S: Fatband representation of the Ni-d and S-p states for both the spin channel.

3 Light hole pocket Heavy hole pocket Electron pocket Figure S3: Fermi surface plot corresponding to the electron and hole pockets. There are two hole pockets around the A (,, ) point. Electron pockets are seen around K (,, ) point and some other points in the BZ. The 3 3 coordinates given in the parenthesis, are in terms of reciprocal lattice vectors. 3

4 Hall coefficient and resistivity The Hall coefficient has been evaluated using the following expression R H = e n lh µ lh +n hhµ hh n eµ e (n lh µ lh +n hh µ hh +n e µ e ) where n lh, n hh, n e are the carrier concentrations of light hole, heavy hole and electron respectively. µ lh, µ hh and µ e are the corresponding mobilities. Since mobility is inversely proportional to the effective mass, the above expression can also be written in the following way R H = e n lh m lh ( n lh m lh + n hh m hh + n hh m hh ne m e + ne m e where m lh, m hh, n e are the effective masses of light hole, heavy hole and electron respectively. To compute the resistivity, the following expression has been used ρ = ) e(n lh µ lh +n hh µ hh +n e µ e ) If we assume that the proportionality constant between the effective mass and mobility as K, the above expression can be written as ρ = ek( n lh + n m hh + ne lh m hh me) We evaluate K by requiring that the calculated resistivity be identical to the experimental resistivity at x =.. This estimated value of K is used to determine the resistivity for all the other compounds with different x-values. 3 Optical Conductivity The optical properties of a material is described by the dielectric function ε(ω) = ε r (ω) + iε i (ω). Interband contribution to the imaginary part of ε(ω) is calculated by convoluting the joint density of states with the dipole matrix elements between the occupied and the unoccupied states. We use the following expression to obtain the imaginary part of the dielectric tensor. ε i αβ(ω) = 4πe dk ψ m ω c,k p α ψ v,k ψ v,k p β ψ c,k δ(ǫ c,k ǫ v,k ω) c,v 4

5 where e is the electronic charge, m is the mass of the electron and ω is the energy of the incident photon. ψ c,k, ψ v,k are the conduction band and valence band wavefunction at point k and ǫ c,k, ǫ v,k are the corresponding energy eigenvalues. Real part of the optical conductivity is given by σ αβ (ω) = ω 4π εi αβ(ω) For the hexagonal system, due to the symmetry, all the off-diagonal components of the optical conductivity are zero and diagonal components obey the relationship: σ xx = σ yy σ zz. The interband part of the optical conductivity is broadened by a Lorentzian function. The broadening parameter of the Lorentzian (Γ) is taken to be linearly dependent with the photon energy, having the form: Γ =.3+.5ω. The intraband part of the optical conductivity (Drude term) is obtained from plasma frequency (ω p ). To compute the plasma frequency, we use the following expression ω p,αα = e m π l dk ψ l,k p α ψ l,k δ(ǫ l,k ǫ F ) where ǫ F is the Fermi energy. Intraband component of the real part of the optical conductivity is given by σ αα (ω) = ω p,αα 4π Γ ω +Γ where Γ is the broadening parameter for the Drude term. Finallyweaddtheintrabandandinterbandpartofeachcomponentofthe optical conductivity and take an average of all the components to compare with the experimental optical conductivity. 4 Various forms of GW calculations We have carried out the various forms of GW calculations as implemented in the VASP code [, ] to show that all important features of the electronic structure of NiS are independent of the specific calculation scheme adopted. In Fig. S4 we have displayed the band dispersion for the low temperature phase of NiS along various high symmetry directions calculated within (a) scgw,(b)scgwand(c)lsda+u(withu =5.eV)approach. ThescGW calculations (Fig. S4-(a)) shows a wide (.36 ev) indirect gap semiconductor/insulator. The scgw calculation yields a significantly reduced gap of 5

6 3 (a) scgw (b) scgw (c) LSDA+U 3 3 Energy (ev) Figure S4: Band dispersion of NiS in the antiferromagnetic insulating regime plotted along various high symmetry directions computed within (a) scgw, (b) scgw, and (c) LSDA+U (with U = 5. ev) methods. The figure shows that the essential qualitative parts of the electronic band dispersions remain identical, irrespective of the calculation method used, indicating NiS to be an indirect gap insulator..68 ev (see Fig. S4-(b)) but all the essential features of band dispersions remain the same as in the case of scgw results (Fig. S4-(a)). The LSDA+U calculations with U = 5. ev, shown in Fig. S4-(c) gives essentially the same band dispersions as in scgw with an identical band gap of.68 ev. We can also obtain a bandgap of.36 ev within LSDA+U with a U of 7. ev. Thus, in the antiferromagnetic insulating regime, the essential qualitative parts of the electronic band dispersions remain identical, irrespective of the calculation method (scgw, scgw or LSDA+U), indicating an indirect gap insulator with very similar dispersions. Therefore, all results, computed on the basis of these results also remain the same; specifically, the experimental optical data, shown as Fig. 3 of our manuscript, remain entirely incompatible with all three results shown in Fig. S4, as the smallest direct gap, defining the smallest energy for optical absorption, are.75,.98, and.9 ev for scgw, scgw and LSDA+U respectively, whereas the experimental value is. ev. Therefore, it is clear that both scgw and scgw are not suitable to describe the low temperature phase of NiS. Next we have performed the scissors operation on the scgw and the scgw band dispersions by vertically moving the empty conduction band down towards the occupied valence band to decrease both the band gap and the optical gap continuously. If we do this to make the bandgap infinitesi- 6

7 (a) scgw (b) scgw (c) LSDA+U Energy (ev).39 ev.3 ev.4 ev Figure S5: Band dispersion for AFM NiS obtained by (a) scgw and (b) scgw, where the band gap is made infinitesimally small using scissors operator. (c) Band dispersion computed within LSDA+U method with U = 3. ev which gives an infinitesimally small band gap. Arrow in each panel indicates the optical gap where in each case it is much larger than the experimental gap of. ev. mally small, we are still left with an optical (direct) gap of.39 and.3 ev for the scgw and scgw, respectively, as shown in Fig. S5-(a) and S5-(b), once again totally inconsistent with experimental finding of a. ev optical gap. Thus, the qualitative features of band dispersions obtained with scgw and scgw already establish that the only way calculated results can be consistent with the experimental results is to push the conduction band further down by about.-.3 ev, making the system metallic, resulting from an indirect overlap of the valence and the conduction bands of these dispersions, independent of the calculation method employed, such that the direct gap can be in the order of. ev required by the experimental result. Avoiding this arbitrary reduction of the bandgap, we can alternately use LSDA+U approach where by tuning U as a parameter, we can reduce the bandgap, exploring whether any self-consistent gapped or insulating band dispersion can be made to agree with the experimental data. Fig. 3 in the manuscript shows that this is not possible, since even an infinitesimally small bandgap (obtained for U = 3. ev), being an indirect one, has a direct (or optical) gap of.4 ev (see Fig. S5-(c)) which is twice as large as what is seen in the experiment, (see Fig. 3 in the manuscript). This shows that independent of the method of calculation, experimental optical data requires the system to be ungapped or metallic, with an indirect overlap of the top 7

8 (a) LSDA+U (b) scgw (c) GW Figure S6: Band dispersion along various high symmetry directions computed within (a) LSDA+U (with U =.8 ev), (b) scgw and (c) GW (on PBE state) approaches for the high temperature Pauli paramagnetic, metallic phase of NiS. of the valence band at the A-point and the bottom of the conduction band at about half-way along the A-L direction. In Fig. S6, we show electron energy dispersions obtained from LDA+U (with U =.8 ev) (Fig. S6-(a)), scgw (Fig. S6-(b)) and GW on PBE (Fig. S6-(c)) for the high temperature Pauli paramagnetic, metallic phase. The comparison in Fig. S6 makes it clear that the three approaches give essentially the same results, once again making our conclusions concerning the high temperature nonmagnetic, metallic state independent of the calculation technique employed. Finally we present the energy dispersions for the antiferromagnetic metallic state within LSDA+U (with U =.3 ev) (Fig. S7-(a)) and GW on PBE (Fig. S7-(b)). Clearly, once again, description of energy dispersions from these two techniques are almost the same, with all important features, discussed in our manuscript, namely self-doping and compensation, being realized in every case. This is a very strong evidence that our novel interpretation of the unusual, low temperature phase of NiS in terms of a novel self-doped, nearly compensated, antiferromagnetic, low-density metal is robust with respect to the specific calculation scheme adopted by us. While we do not have the possibility of calculating the matrix elements relevant for the optical conductivity data within any of the GW approaches, we note that the joint density of states, when weighted by the matrix elements, yields the optical data. Thus, one expects that the qualitative features 8

9 (a) LSDA+U (b) GW Figure S7: Band dispersion of NiS along various high symmetry directions computed within (a) LSDA+U (with U =.3 ev), and (b) GW on PBE approaches for the low temperature antiferromagnetic metallic state. All the calculations capture the important features of the band dispersions, namely the self-doping and compensation as discussed in the text. of the optical data to be reasonably well captured by the energy distribution of the joint density of states. So we have computed the joint density of sates based on the electronic structure results from the GW on PBE calculation of the low temperature crystal structure of NiS; the result is compared with the experimental data along with the calculated LSDA+U results in Fig. S8. Thus, the GW result, without any adjustable parameter and yielding the self-doped metallic state, provides a good description of the experimental data, as shown in Fig. S8, and corresponds closely to the calculated result from LSDA+U approach (which we use for all results in our manuscript), thereby once again establishing the only possible description of the ground state of the low temperature NiS is in terms of the novel state proposed by us and this conclusion is truly independent of the calculation methodology. 9

10 Re [σ] ( 3 Ω -Cm - ).5.5 Experiment U =.3 ev GW Photon Energy (ev) Figure S8: A comparison of the GW optical conductivity with the experimental data and the calculated LSDA+U results (with U =.3 ev). The good agreement of GW with the experimental results and LSDA+U approach establish the robustness of our conclusion as discussed in the text. References [] Kresse, G., & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, (993). [] Kresse, G., & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, (996).